I think you can use functions from rsconnect
to deploy to shinyapps.io.
Admin docs explains it : http://docs.rstudio.com/shinyapps.io/getting-started.html#deploying-applications
using a Continuons Integration tool or service (Travis, Jenkins, GitlabCI, ...), you can automate the deployement. The tooling depends on your technical preference and environment. Travis is widely use in the open source community. At work, I have Jenkins.
There are some example of that for RStudio Connect product from where you could get inspiration
http://docs.rstudio.com/connect/admin/appendix-ci.html
I should precise that I have never done it myself on shinyapps.io but I think it is feasible. Maybe someone will have an example...